Eventual periodicity of the Smith forms of integer matrix powers

TL;DR

This paper proves that the Smith forms of powers of integer matrices eventually become periodic up to a constant diagonal factor, confirming a 2013 conjecture and revealing that the starting point and period can be arbitrarily large.

Contribution

It establishes the eventual periodicity of Smith forms of integer matrix powers and shows that the initial index and period can be arbitrarily large, confirming a longstanding conjecture.

Findings

Smith forms of matrix powers are eventually periodic.

The initial index and period can be arbitrarily large.

Provides an affirmative answer to a 2013 conjecture.

Abstract

We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if $\mathrm{SF}(M)$ denotes the Smith form of $M \in \Z^{m \times m}$, then for every $A \in \Z^{m \times m}$ there exist $n_0 \in \N$, an integer $T \geq 1$, and a constant diagonal matrix $D \in \Z^{m \times m}$ such that $n \geq n_0$ implies $\mathrm{SF}(A^{n+T})=D \cdot \mathrm{SF}(A^n)$. This provides an eventually affirmative answer to a conjecture posed in 2013 by R. Bruner. We also show that both $n_0$ and $T$ can be arbitrarily large.